Not long ago, President Donald Trump’s advisers believed he would soon fire the Justice Department official overseeing a Russia investigation he calls a “witch hunt,” a step that could set off a cascade of resignations and a potential crisis.

But in recent days a series of events have played out that have calmed the situation, easing fears that Mr. Trump would imminently move against Deputy Attorney General Rod Rosenstein.

In a private meeting at the White House on April 12, Mr. Rosenstein told the president he wasn’t a target of the Russia investigation. During the meeting, Mr. Rosenstein left the impression that the president was “in no jeopardy,” as one senior White House official briefed on the exchange described it. It is unclear exactly what was said in the room.

Telling a person he is not a target of a probe doesn’t mean that prosecutors have concluded their investigation, but rather that they don’t have sufficient evidence at the time tying the person to a crime.

Mr. Trump last week added to his legal team former New York City Mayor and U.S. Attorney Rudy Giuliani, whose self-described mission is bringing the investigation to a quick close—a goal that has eluded the Trump legal team and remains important to the president. It is not yet clear how Mr. Giuliani intends to do that, and Special Counsel Robert Mueller has given no indication he would conclude the investigation before he has explored every avenue he deems necessary.

On Sunday, White House legislative affairs director Marc Short said on NBC’s “Meet the Press” of Messrs. Rosenstein and Mueller, “As far as I know, the president has no intention of firing these individuals.”

Asked why the president wouldn’t definitively lay to rest speculation that he was considering the dismissals, Mr. Short said, “Because you don’t know how far off this investigation is going to veer. Right now, he has no intention of firing him.”

In another development that potentially solidifies Mr. Rosenstein’s position, Attorney General Jeff Sessions warned White House counsel Donald McGahn last week that he would consider quitting if Mr. Trump were to fire his deputy.

Mr. Trump himself addressed the investigation on Sunday, tweeting in response to reports that Republican lawmakers were asking Mr. Sessions to investigate former FBI Director James Comey and former Democratic presidential candidate Hillary Clinton, “Good luck with that request!”

While Mr. Trump has made plain his frustration with Mr. Sessions for recusing himself from the Russia probe, White House officials have repeatedly advised him that the attorney general should stay on the job and that pushing him out would create a messy backlash and prolong the investigation.

Over the past year, Mr. Trump has been apt to change his mind when it comes to personnel, souring on officials who had seemed secure and retaining those whose position appeared shaky.

But White House officials now believe the president won’t dismiss Messrs. Rosenstein or Mueller anytime soon. Asked on Wednesday whether he might do so, Mr. Trump dismissed the notion by saying that the rumor had swirled for months, but “they’re still here.”

Lawmakers are considering legislation aimed at protecting Mr. Mueller’s investigation from White House interference. This week, the Senate Judiciary Committee is set to vote on a bill that would prevent Mr. Mueller from being fired without good cause while the Russia investigation continues.

Sen. Susan Collins (R., Maine) defended the Mueller investigation on Sunday. “I believe he’s staying within the parameters.” she said in on NBC.

Mr. Mueller is examining whether Trump associates colluded with Russia’s efforts to interfere in the 2016 U.S. election and whether Mr. Trump obstructed justice. Mr. Trump has denied both collusion and obstruction, and Moscow has denied election meddling.

The landscape is notably different from earlier this month. Mr. Trump was furious about an April 9 FBI raid targeting his lawyer Michael Cohen, an action that Mr. Rosenstein had approved. Two days later, Mr. Trump attacked Mr. Rosenstein on Twitter for the first time and took aim at Mr. Mueller, whom he accused of being “conflicted.” People who had spoken to Mr. Trump last week came away convinced that Mr. Rosenstein was on his way out.

Mr. Cohen is under criminal investigation for potential bank fraud and campaign-finance violations. He has denied wrongdoing.

Helping distract from Messrs. Rosenstein and Mueller in recent days has been the public re-emergence of James Comey. The president last week trained his ire on the former FBI director, who released a book about his dealings with the president and has conducted several TV interviews, including one in which he called Mr. Trump “morally unfit” to serve as president.

Over the past week, Mr. Trump has issued a dozen tweets about the former FBI director, calling him “shady,” “slippery,” “third-rate,” and “out of whack.” He has made no mention on Twitter of Mr. Rosenstein.

An open question is whether Mr. Trump will agree to an interview with Mr. Mueller. He has said he is willing to do so, though he also has expressed mounting disdain for the investigation.

Late Friday night, he suggested in a tweet that the investigation is tainted given actions taken by Mr. Comey to trigger the appointment of a special counsel. He wrote that Mr. Comey had “illegally” passed classified information to reporters.

“Therefore,” he wrote, “the Special Council was established based on an illegal act? Really, does everybody know what that means?”

Last year, Mr. Comey told a Senate committee that he arranged to leak information to a reporter about conversations with Mr. Trump in hopes of spurring the appointment of a special counsel—a gambit that proved successful.